re: Is it possible, with these short codes to create additional drop downs for these categories?
Sorry, no, there is not.
There’s a solution like this one:
It is a custom developed solution for them. I have further developed it so it can be used on any site but at this point still requires me to install and configure which is a service I do provide if you are interested. You would basically have a setup like theirs. You doctor page would be this search, after the user enters their search criteria, they will be taken to the search results page.